A. Fonda and R. Toader - Dipartimento di Matematica e Informatica

J. Differential Equations 244 (2008) 3235–3264 www.elsevier.com/locate/jde

Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach Alessandro Fonda a,∗ , Rodica Toader b a Dipartimento di Matematica e Informatica, Università di Trieste, P.le Europa 1, I-34127 Trieste, Italy b Dipartimento di Ingegneria Civile, Università di Udine, Via delle Scienze 208, I-33100 Udine, Italy

Received 3 July 2007; revised 5 November 2007 Available online 8 January 2008

Abstract We are concerned with non-autonomous radially symmetric systems with a singularity, which are T -periodic in time. By the use of topological degree theory, we prove the existence of large-amplitude periodic solutions whose minimal period is an integer multiple of T . Precise estimates are then given in the case of Keplerian-like systems, showing some resemblance between the orbits of those solutions and the circular orbits of the corresponding classical autonomous system. © 2007 Elsevier Inc. All rights reserved. MSC: 34C25 Keywords: Periodic solutions; Nonlinear dynamics

1. Introduction In this paper, we are mainly interested in the study of periodic solutions to systems of the type x¨ + f t, |x| x = 0.

(1)

Here, f is a real function, T -periodic in t, which is defined on R × ]0, +∞[, so that 0 can be a singularity. We therefore look for solutions x(t) ∈ RN which never attain the singularity, in the sense that * Corresponding author.

E-mail addresses: [email protected] (A. Fonda), [email protected] (R. Toader). 0022-0396/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jde.2007.11.005

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A. Fonda, R. Toader / J. Differential Equations 244 (2008) 3235–3264

x(t) = 0,

for every t ∈ R.

(2)

We would like to deal, for example, with equations like x c(t) x = e(t) , γ |x| |x|

(3)

c(t) d(t) x x+ x = e(t) . 3 4 |x| |x| |x|

(4)

x¨ + or x¨ +

In the above, c(t), d(t) and e(t) are T -periodic functions, and γ is a positive constant. Notice that, if c, d, e are constant functions, with e(t) identically zero, and γ = 3, then (3) is the equation of the Keplerian motion of a planet, while (4) is, in the simplest approximated form, the equation of its relativistic motion (cf. [24, p. 122]). Since (1) is radially symmetric, its solutions have planar orbits. This fact simplifies considerably the analysis, and plays a crucial role in our approach. More general systems, of the type x¨ + ∇V (t, x) = 0,

(5)

were studied by many authors, mainly by the use of variational methods, assuming the function V to be T -periodic in t, differentiable in x = 0 with continuous gradient, and such that lim V (t, x) = −∞.

x→0

(6)

In the planar case, Gordon [18] was able to prove a nice existence result for periodic solutions of (5) by the use of a variational method. The main difficulty of avoiding collision orbits (for which (2) is not satisfied) was overcome by introducing the so-called “strong force” assumption. His result, later improved by Capozzi, Greco and Salvatore [10], can be stated for our convenience as follows. Theorem 1. Let N = 2, assume (6) and that there exist ν ∈ [0, 2[ and positive constants c1 , c2 such that V (t, x) c1 |x|ν + c2 , for every t and x = 0. If moreover (SF) there are a C 1 -function U : R2 \ {0} → R, a neighborhood N of 0 and a positive constant c3 such that limx→0 U (x) = −∞ and 2 −V (t, x) ∇U (x) − c3 ,

for every x ∈ N \ {0},

then, for every integer k 1, system (5) has a periodic solution with minimal period kT .


3237

The proof, originally given only for period T , consists in minimizing the associated functional over the orbits which turn around the origin exactly once in their period time. The strong force condition (SF) guarantees that the minimization procedure does not lead to a collision orbit. Theorem 1 applies to Eq. (3) provided that γ 4, and does not apply to Eq. (4), since condition (SF) does not hold there. After the pioneering paper by Gordon, much attention was given to systems in higher dimensions (see, e.g., [2,3,7,19–21,25,26]), and trying to avoid the strong force assumption (see, e.g., [12,13,29,31]). The possibility of obtaining collision orbits in situations when the strong force condition is not satisfied led Bahri and Rabinowitz [7] to the definition of some kind of “generalized solution.” Many authors then studied this type of “solutions” (see [5] and the references therein). However, we will never consider such a situation, and for a solution we will always ask that (2) holds. We will be mainly interested in situations which include the Keplerian case, where the strong force assumption does not hold. A remarkable result in this direction was obtained by Degiovanni, Giannoni and Marino [14]. In their paper, a lower semicontinuous function φ : [1, +∞[ → [1, +∞] is defined, with the following property: φ(α) = 1,

if α = 1,

φ(α) ∈ ]1, +∞[,

if 1 < α < 2,

φ(α) = +∞,

if α 2.

This function was recalled in [15], where Degiovanni and Giannoni, developing the ideas in [14], proposed the following result. Theorem 2. Let V be even in x and assume that, for some α 1, c c −V (t, x) α , α |x| |x|

(7)

for every t and x = 0. If 0 < c 2α φ(α)c, then, for every integer k 1, system (5) has a periodic solution xk (t) having period kT , whose orbit is symmetric with respect to the origin, and lim min|xk | = +∞.

k→∞

Theorem 2 holds in any dimension N and the function V is not required to be radially symmetric. The minimality of the period kT is not guaranteed, but there surely are infinitely many periodic solutions. The result applies to Eqs. (3) and (4) if e(t) is identically zero and γ = α + 2 3, with suitable bounds on c(t) when 3 γ < 4. Conditions similar to (7) were proposed by Ramos and Terracini [27] in the case 0 < α < 1, while considering autonomous equations. The Keplerian exponent α = 1 proved to be particularly delicate when dealing with such methods, as shown by Capozzi, Solimini and Terracini [11].

3238


A different approach was used by Ambrosetti and Coti Zelati in [4], where a perturbative situation of the type V (t, x) = −

c + εW (t, x) |x|α

was considered. Here c > 0 is a constant, α is positive, and ε is sufficiently small. They found several periodic solutions “bifurcating” from the circular orbits of the unperturbed system. Their results apply to Eq. (3), when c(t) is constant and e(t) is sufficiently small. Another result which includes the Keplerian case can be found in [28], where Serra and Terracini considered C 2 -potentials of the type V (t, x) = −

c λ + |x|2 + W (t, x), α |x| 2

where c > 0 is a constant, α ∈ ]0, 2[ and W includes terms of lower order near 0 (see their assumptions (H1) and (H2)), thus proving the following. Theorem 3. Let N 3 and assume V as above, with W such that T W t + , −x = W (t, x), 2 for every t, x = 0, and lim

|x|→∞

∇W (t, x) =0 |x|

uniformly in t. If λ < ( Tπ )2 , then (5) has a T -periodic solution, whose orbit is symmetric with respect to the origin. This result applies (with λ = 0) to Eq. (3) whenever c(t) is constant, 2 < γ < 4, and e(t) is twice continuously differentiable. See also [1] where, under similar hypotheses, a different type of singularity is considered, and the existence of a T -periodic solution is proved. We deal here with an equation like (1), which for convenience we write as x . x¨ = −h t, |x| + e(t) |x|

(8)

We assume that h : R × ]0, +∞[ → R is L1 -Carathéodory and T -periodic with respect to its first variable, and that e ∈ L1loc (R) is T -periodic, with

(H0)

1 e := T

T e(t) dt 0. 0

Let us state our main existence result.


3239

Theorem 4. Assume (H0) and the following hypotheses: (H1)

lim r 3 h(t, r) = +∞,

r→+∞

h(t, r) =0 r→+∞ r

(H2)

lim

uniformly for almost every t. Then, there exists k1 1 such that, for every integer k k1 , Eq. (8) has a periodic solution xk (t) with minimal period kT , which makes exactly one revolution around the origin in the period time kT . Moreover, lim min|xk | = +∞

k→∞

and lim

k→∞

min|xk | = 1. max|xk |

Notice that we have no assumption on h near the origin. The orbits of the solutions we find are close to be circular, in the sense that the quotient between the minimal and the maximal radius is close to 1, and stay far away from the origin. This fact can be used to deal with more general equations, like (5), when the potential V is assumed to be radially symmetric only outside a given compact set. Our result directly applies to (3) and (4), as the following two corollaries show. Corollary 1. Assume (H0), let c ∈ L∞ (R) be such that (H3)

0 < c1 c(t) c2 ,

for some positive constants c1 and c2 , and let 0 < γ < 4. Then, the conclusion of Theorem 4 holds for Eq. (3). Notice that, when 0 < γ < 1, there is no singularity at all. However, the fact that γ has to be nonzero is crucial in order to avoid the possibility of linear resonance phenomena. Corollary 2. Assume (H0), (H3), and let d ∈ L∞ (R). Then, the conclusion of Theorem 4 holds for Eq. (4). The proof of Theorem 4 is given in the next section. It is based on a careful analysis of the behaviour of the solutions having large angular momentum. The radial component of our solutions is T -periodic and its oscillations are controlled in some way by its amplitude, while the angle component varies very slowly, and can be tuned so that the solution performs exactly one rotation in a period kT , for k large enough. In Section 3 we will give more precise estimates on the solutions and their velocities for equations like (3) and (4). The resemblance with the circular orbits of the classical autonomous equations will then be even more evident.

3240


In Section 4, we will discuss the possibility of obtaining multiplicity of solutions with the same minimal period, distinguishing them by the number of rotations they perform in their period time. In Section 5 we show that the conclusion of Theorem 4 cannot be true without the assumption (H0). Indeed, we prove that, if e > 0 and h(t, r) tends to zero as r tends to infinity, there is a compact region with the property that all solutions exiting this region necessarily must have unbounded orbits. In Appendix A we give an elementary proof of the fact that the solutions of a radially symmetric system like (1) have planar orbits. This is well known in the case N = 3, but it seems hard to find it proved, for higher dimensions, in standard textbooks. 2. Proof of the main result This section is dedicated to the proof of Theorem 4. Let us first clarify our assumptions. Recall that L1 -Carathéodory means • h(·, r) is measurable, for every r > 0; • h(t, ·) is continuous, for a.e. t ∈ [0, T ]; • for every compact interval [a, b] in ]0, +∞[, there exists a,b ∈ L1 (0, T ) such that r ∈ [a, b]

⇒

h(t, r) a,b (t),

for a.e. t ∈ [0, T ].

Since system (8) is radially symmetric, the orbit of a solution always lies on a plane containing the origin (see Appendix A). Moreover, since rotations leaving fixed the origin preserve the solutions, it is enough to look for those solutions whose orbits lie on a fixed plane. Therefore, from now on, we will assume, without loss of generality, that N = 2. We may write the solutions in polar coordinates: x(t) = ρ(t) cos ϕ(t), sin ϕ(t) ,

(9)

and (2) is satisfied if ρ(t) > 0, for every t. Eq. (8) is then equivalent to the system

(S)

⎧ 2 ⎪ ⎨ ρ¨ − μ + h(t, ρ) = e(t), ρ3 ⎪ ⎩ 2 ρ ϕ˙ = μ,

where μ is the (scalar) angular momentum of x(t), cf. [6]. Recall that μ is constant in time along any solution. In the following, when considering a solution of (S), we will always implicitly assume that ρ > 0. Let us define the modified function h˜ : R × ]0, +∞[ → R as follows: ⎧ h(t, r), ⎪ ⎨ ˜ r) = (2r − 1)h(t, r), h(t, ⎪ ⎩ 0,

if r 1, if

1 2

r 1,

if 0 < r 12 .


3241

This function still satisfies the L1 -Carathéodory conditions, as well as assumptions (H1) and (H2). Correspondingly, let us consider the modified equation x x¨ = −h˜ t, |x| + e(t) , |x|

(10)

which is equivalent to the system ⎧ 2 ⎪ ⎨ ρ¨ − μ + h(t, ˜ ρ) = e(t), 3 ρ ⎪ ⎩ 2 ρ ϕ˙ = μ.

˜ (S)

We will eventually find solutions of (10) which do not enter the circle of radius 1, so that along these solutions the functions h and h˜ coincide. ˜ has a T -periodic We begin by showing that, for μ large enough, the first equation in (S) solution. To this aim, we use degree theory. For λ ∈ [0, 1], we define ˜ r) + 1 − λ , hλ (t, r) = λh(t, r2

eλ (t) = λe(t) + (1 − λ)e,

and consider the equation ρ¨ −

μ2 + hλ (t, ρ) = eλ (t). ρ3

(11)

Notice that the hypotheses (H1) and (H2) are also satisfied by hλ and, moreover, the L1 Carathéodory conditions hold true. In particular, • for every compact interval [a, b] in ]0, +∞[ there exists ã,b ∈ L1 (0, T ) such that r ∈ [a, b] ⇒ hλ (t, r) ã,b (t), for a.e. t ∈ [0, T ] and every λ ∈ [0, 1]. Lemma 1. For every Γ > 0 there exists μ(Γ ) 1 such that, if μ μ(Γ ), λ ∈ [0, 1], and ρ is a T -periodic solution of (11), then ρ ∞ Γ . Proof. Assume by contradiction that there are Γ > 0 and sequences (λn )n , (μn )n , and (ρn )n such that λn ∈ [0, 1], limn μn = +∞, and ρn is a T -periodic solution of (11) for λ = λn and μ = μn , with ρn ∞ < Γ . Multiplying in (11) by ρn3 and integrating gives T −3

T ρ˙n2 ρn2

0

− μ2n T

+

T hλn (t, ρn )ρn3

0

=

eλn ρn3 , 0

so that, since eλn 1 e 1 , T hλn (t, ρn )ρn3 μ2n T − e 1 Γ 3 . 0

3242


On the other hand,

T

hλn (t, ρn )ρn3 = 0

hλn (t, ρn )ρn3 +

{0 1 such that, if μ ∈ [A, B], λ ∈ [0, 1], and ρ is a T -periodic solution of (11), then ρ(t) ˙ < C,

1 < ρ(t) < C, for every t ∈ [0, T ].

Proof. By the choice of μ and Lemma 1, it has to be ρ ∞ Γ , and by Lemma 2, 1 1 min ρ ρ ∞ − C Γ − C > 1. 2 2 Using (H1), let us choose r˜ > 0 such that r r˜

⇒

r 3 hλ (t, r) 2B 2 ,

for a.e. t ∈ [0, T ] and every λ ∈ [0, 1].

Let us first show that min ρ < r˜ . Integrating (11), using (H0) we have T 0

μ2 = ρ3

T

T hλ (t, ρ) − T e

0

hλ (t, ρ). 0

Arguing by contradiction, if min ρ r˜ , the previous inequality would give T 0

μ2 ρ3

T

2B 2 , ρ3

0

which is impossible, since μ ∈ [A, B]. Having proved that min ρ < r˜ , by Lemma 2 we deduce that

ρ ∞ 2(min ρ + C ) < 2(˜r + C ).

3244


Hence, setting b = 2(˜r + C ), from Eq. (11) and the Carathéodory conditions, since ρ(t) ˙ vanishes somewhere, T

ρ

˙ ∞ ρ

¨ 1 0

μ2 + ρ3

T

hλ (t, ρ) + eλ 1 < B 2 T + ˜1,b 1 + e 1 := b .

0

Defining C = max{b, b }, the proof is completed.

2

As a consequence of Lemma 3, if μ μ and ρ(t) is a T -periodic solution of ρ¨ −

μ2 ˜ + h(t, ρ) = e(t), ρ3

(12)

˜ then ρ(t) also satisfies the first equation in (S), i.e. which is the first equation in (S), ρ¨ −

μ2 + h(t, ρ) = e(t). ρ3

(13)

The following lemma gives us an important information concerning this equation. Let us denote by CT1 the set of T -periodic C 1 -functions, with the usual norm of C 1 ([0, T ]). Lemma 4. Given A, B, with μ A B, there is a continuum CA,B in [A, B] × CT1 , connecting {A} × CT1 with {B} × CT1 , whose elements (μ, ρ) are solutions of both Eqs. (12) and (13). Proof. By the above remark, it is enough to prove the lemma for the modified equation (12). In order to apply degree theory, let us define the following operators: L : D(L) ⊂ C 1 [0, T ] → L1 (0, T ),

˙ = ρ(T ˙ ) , D(L) = ρ ∈ W 2,1 (0, T ): ρ(0) = ρ(T ), ρ(0) Lρ = ρ¨ − ρ, and N : [A, B] × C 1 [0, T ] → L1 (0, T ), N (μ, ρ)(t) =

μ2 − h˜ t, ρ(t) + e(t) − ρ(t). 3 ρ (t)

The T -periodic problem for Eq. (12) is thus equivalent to Lρ = N (μ, ρ). Since L is invertible, we can write equivalently ρ − L−1 N (μ, ρ) = 0.

(14)


3245

Let C > 0 be the constant given by Lemma 3 and define G to be the following open subset of C 1 ([0, T ]):

˙ < C, for every t ∈ [0, T ] . G = ρ ∈ C 1 [0, T ] : 1 < ρ(t) < C and ρ(t) By Lemma 3, Eq. (14) has no solutions (μ, ρ) on [A, B] × ∂G. Since L−1 N (μ, ·) is a compact operator, by the global continuation principle of Leray–Schauder (see, e.g., [32, Theorem 14.C]), the lemma will be proved if we show that the degree is nonzero for some μ ∈ [A, B]. In order to compute the degree, we consider Eq. (11). By Lemma 3, the degree has to be the same for every λ ∈ [0, 1]. Let us then take λ = 0, so that (11) becomes ρ¨ −

1 μ2 + = e. ρ3 ρ2

Define the function F : [1, C] × [−C, C] → R2 , 1 μ2 F (u, v) = v, 3 − 2 + e . u u By a result of Capietto, Mawhin and Zanolin [9, Theorem 1], one can compute the Leray– Schauder degree of I − L−1 N (μ, ·) as the Brouwer degree of F : dLS I − L−1 N (μ, ·), G = dB F, ]1, C[ × ]−C, C[ . Since F has a unique zero (u0 , v0 ), in the set ]1, C[ × ]−C, C[, and the Jacobian matrix JF (u0 , v0 ) has a positive determinant, we conclude that the degree has to be equal to 1. 2 We have thus showed that, for every μ μ, Eq. (13) has a T -periodic solution. Let us mention that, following [23], many authors have considered scalar equations with a singularity, like (13). A similar existence result was proved, e.g., in [8], via lower and upper solutions techniques. See also, among others, [16,17,22,30]. We can deduce from Lemma 4 that there is a connected set C, contained in [μ, +∞[×CT1 , which connects {μ} × CT1 with {μ∗ } × CT1 , for every μ∗ > μ, whose elements (μ, ρ) are solutions of both Eqs. (12) and (13). Lemma 5. For every ε > 0, there exists με μ such that, if (μ, ρ) ∈ C with μ με , then T

μ ρ 2 (t)

dt ε.

0

Proof. Given ε > 0, set ε :=

ε 36T

2 .

3246


Let C be as in Lemma 2 and recall that e 0. By (H2), there exists r max{−2e/ε , 3C} such that r r

⇒

h(t, ˜ r) ε r, 2

for a.e. t ∈ [0, T ].

For Γ := 2(r + C ), let μ(Γ ) be as in Lemma 1. Set με := max{μ(Γ ), μ}. Let (μ, ρ) be an element of C, with μ με . By Lemma 1, ρ ∞ Γ , and by Lemma 2, ρ(t) 12 ρ ∞ − C r , for every t ∈ R. Hence, 1 T

T

˜ ρ) 1 h(t, T

0

T

ε ε ρ ρ ∞ . 2 2

0

Integrating in (12), we have 1 T

T

μ2 1 = 3 T ρ

0

T

˜ ρ) − e ε ρ ∞ − e. h(t, 2

0

On the other hand, 1 T

T 0

μ2 μ2 , ρ3

ρ 3∞

so that μ2 e e ε ε − − ε . 2

ρ ∞ 2 r

ρ 4∞ Then, using again Lemma 2, T

μ ρ 2 (t)

dt T

0

μ 4μ T 2 (min ρ) ( ρ ∞ − 2C )2

2 2 √ 2 ρ ∞ 2r μ T ε

ρ 2∞ ρ ∞ − 2C r − 2C 2 √ √ 6C T ε = 36T ε = ε, 3C − 2C =T

thus proving the lemma.

2


3247

Since the function T (μ, ρ) →

μ ρ 2 (t)

dt

0

is continuous from C to R, and C is connected, its image is an interval. By Lemmas 4 and 5 this interval is of the type ]0, θ ] for some θ > 0. Lemma 6. For every θ ∈ ]0, θ ], there are (μ, ρ, ϕ), verifying system (S), for which (μ, ρ) ∈ C, and ρ(t + T ) = ρ(t),

ϕ(t + T ) = ϕ(t) + θ,

for every t ∈ R. Proof. Given θ ∈ ]0, θ ], there are (μ, ρ) ∈ C such that T

μ dt = θ. ρ 2 (t)

0

In particular, the first equation in (S) is satisfied and ρ is T -periodic. Moreover, defining t ϕ(t) =

μ ρ 2 (s)

ds,

0

the second equation in (S) is also satisfied and t+T

ϕ(t + T ) − ϕ(t) = t

μ ds = 2 ρ (s)

T

μ ρ 2 (s)

ds = θ.

2

0

We are going to complete now the proof of Theorem 4. For every θ ∈ ]0, θ ], the solution of system (S) found in Lemma 6 provides, through (9), a solution to Eq. (8) such that x(t + T ) = eiθ x(t), for every t ∈ R. In particular, if θ = 2π k for some integer k 1, then x(t) is periodic with minimal period kT , and rotates exactly once around the origin in the period time kT . Hence, for every integer k 2π/θ , we have such a kT -periodic solution, which we denote by xk (t). Let (ρk (t), ϕk (t)) be

3248


its polar coordinates, and μk be its angular momentum. By the above construction, (μk , ρk , ϕk ) verify system (S), (μk , ρk ) ∈ C, and T 0

2π μk dt = . 2 k ρk (t)

(15)

We claim that lim μk = +∞.

(16)

k→∞

Indeed, if (μkj )j were a bounded subsequence, with μkj ∈ [μ, B] for some B, using Lemma 3 with λ = 1, there would be a constant C > 0 such that ρkj ∞ < C, and hence T 0

μkj ρk2j (t)

dt >

μT , C2

for every j , in contradiction with (15). By (16) and Lemma 1, with λ = 1, we have lim ρk ∞ = +∞.

k→∞

By Lemma 2, again with λ = 1, lim (min ρk ) = +∞.

(17)

k→∞

Using (H2), for any fixed ε > 0 there is rε > 0 such that r rε ⇒ h(t, r) εr, for a.e. t ∈ [0, T ]. By (17), there is kε > 0 such that ⇒

k kε

e 1 , ρk (t) max rε , εT

for every t, so that, from the first equation in (S),

ρ˙k 22

T T μ2 − 2 + h(t, ρk )ρk − eρk = ρk 0

0

T

T h(t, ρk )ρk −

0

eρk 0

εT ρk 2∞ + e 1 ρk ∞ 2εT ρk 2∞ .


3249

Hence, √ max ρk − min ρk T ρ˙k 2 √ 2εT . max ρk

ρk ∞ The above proves that max ρk − min ρk = 0, max ρk

lim

k→∞

(18)

thus concluding the proof of Theorem 4. 3. Applications to Keplerian-like systems In this section we study more in detail equations like (3) and (4). To this aim, let us state the following direct consequence of Theorem 4. Corollary 3. Assume (H0), and that there is γ ∈ ]0, 4[ such that (H4)

0 < c1 lim inf r γ −1 h(t, r) lim sup r γ −1 h(t, r) c2 r→+∞

r→+∞

uniformly for almost every t. Then, there exists k1 1 such that, for every integer k k1 , Eq. (8) has a periodic solution xk (t) with minimal period kT , which makes exactly one revolution around the origin in the period time kT . Moreover, lim min|xk | = +∞

k→∞

and min|xk | = 1. k→∞ max|xk | lim

Notice that Corollaries 1 and 2 follow directly from Corollary 3. Our aim will be now to give more precise estimates on the amplitude of the solutions having large minimal period, and on their radial and tangential speed. The radial and tangential velocities of x(t) are, as well known, defined by x˙ rad =

x˙ · x x, |x|2

x˙ tan = x˙ − x˙ rad .

Recall that, passing to polar coordinates, Eq. (8) is equivalent to (S). Lemma 7. Assume (H0) and (H4), with 0 < γ < 4. For every ε ∈ ]0, 1[ there is με > 0 such that, if μ με and ρ is a T -periodic solution of the first equation in (S), then, for every t,

3250


• when, either e < 0 and 0 < γ < 1, or e = 0, one has 2 1 2 1 μ 4−γ μ 4−γ ρ(t) (1 + ε) ; (1 − ε) c2 c1 • when e 0 and γ = 1, it is 1 1 3 3 μ2 μ2 ρ(t) (1 + ε) ; (1 − ε) c2 − e c1 − e • when e < 0 and 1 < γ < 4, 2 1 2 1 μ 3 μ 3 ρ(t) (1 + ε) ; (1 − ε) −e −e • finally, when e 0 and 1 γ < 4, we have ρ(t) ˙ e 1 + ε.

(19)

Proof. Let c1 and c2 be such that 0 < c1 < c1 c2 < c2 .

(20)

By (H4), there is r∗ such that ⇒

r r∗

c1

r γ −1

h(t, r)

c2

r γ −1

for a.e. t ∈ [0, T ].

,

(21)

From Lemmas 1 and 2 we know that, for sufficiently large μ, it has to be ρ(t) r∗ , for every t. Let us consider the function gμ,c : ]0, +∞[ → R defined by gμ,c (r) = −

μ2 c + γ −1 . 3 r r

(22)

Note that the equation gμ,c (r) = e has a unique solution, which we denote by r μ,c : gμ,c (r μ,c ) = e. Integrating in (13) and using (21), we get 1 T

T 0

c μ2 − 3 + γ 1−1 ρ ρ

1 e T

T

c2 μ2 − 3 + γ −1 . ρ ρ

0

Hence, there exist t 1 , t 2 ∈ [0, T ] such that ρ(t 1 ) r μ,c and ρ(t 2 ) r μ,c . As r μ,c r μ,c , we 1 2 2 1 have that there exists t ∈ [0, T ] such that ρ(t) ∈ [r μ,c2 , r μ,c1 ].

(23)


3251

Let us now give some estimates on r μ,c . It is easy to see that lim r μ,c = +∞.

μ→+∞

If e = 0, we immediately have r μ,c =

μ2 c

1 4−γ

.

If e < 0 and 0 < γ < 1, we do not have the equality, but nevertheless, using (24), lim

r μ,c

1 μ→+∞ μ2 4−γ (c)

= 1.

If e 0 and γ = 1, it is easy to compute r μ,c =

μ2 c−e

1 3

.

Finally, if e < 0 and 1 < γ < 4, let us fix ε ∈ ]0, 1[. Since

γ −1 2 1 e μ 3 c(−e) 3 = + , gμ,c (1 − ε ) −e (1 − ε )3 (1 − ε )γ −1 μ 2(γ3−1) γ −1 2 1 e μ 3 c(−e) 3 = + , gμ,c (1 + ε ) −e (1 + ε )3 (1 + ε )γ −1 μ 2(γ3−1)

for μ sufficiently large we have 2 1 2 1 μ 3 μ 3 < e < gμ,c (1 + ε ) , gμ,c (1 − ε ) −e −e hence 2 1 2 1 μ 3 μ 3 (1 − ε ) < r μ,c < (1 + ε ) . −e −e

The above proves that, in this case, lim

μ→+∞

r μ,c 2

1

μ 3 ( −e )

= 1.

As already seen in the previous section, if μ is sufficiently large, we have min ρ max ρ − 1 < ε.

(24)

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Hence, (1 − ε)ρ(t) ρ(t) (1 + ε)ρ(t), for every t, and the above estimates on ρ(t), together with the arbitrary choice of c1 , c2 satisfying (20), lead to all the claimed estimates for ρ(t). Notice that, if 1 γ < 4, the function gμ,c defined in the proof of Lemma 7 is bounded above: γ −1 3 c 4−γ γ − 1 4−γ gμ,c (r) (4 − γ ) := Mμ,c , 3 μ2 and lim Mμ,c = 0.

μ→+∞

Hence, for μ sufficiently large, we have 1 ρ(t) ¨ −gμ,c2 ρ(t) + e(t) −Mμ,c2 + e(t) − ε + e(t). T So, for every t1 , t2 in [0, T ], t2 ρ(t ˙ 2 ) − ρ(t ˙ 1) =

ρ(t) ¨ dt −ε − e 1 , t1

from which we deduce, since ρ˙ vanishes somewhere, that ρ(t) ˙ ε + e 1 , for every t.

2

We will now concentrate on the case e = 0. Theorem 5. Assume (H4), with 0 < γ < 4. If e = 0, then the solutions xk (t) of Eq. (8) obtained in Corollary 3 satisfy the following estimates: 1 γ4 −1 1 |xk (t)| |xk (t)| γ4 −1 4−γ c1 c2 4−γ lim inf lim sup c2 c1 , 2 2 k→∞ kT γ k→∞ ( kT ) γ ( 2π ) 2π

(25)

8−γ −4 1 1 |x˙ tan (t)| |x˙ tan (t)| 8−γ c1 c2 γ (4−γ ) lim inf k 2−γ lim sup k 2−γ c2 c1−4 γ (4−γ ) , k→∞ kT k→∞ ( kT ) γ ( 2π ) γ 2π

(26)

and moreover, if 1 γ < 4, lim supx˙krad (t) e 1 k→∞

uniformly in t.

(27)


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Proof. Let μk , ρk (t) and ϕk (t) be the angular momentum and the polar coordinates corresponding to xk (t), respectively. Fix ε ∈ ]0, 1[. Using Lemma 7 and (16), for k sufficiently large we can write 2 1 2 1 μ 4−γ μ 4−γ ρk (t) (1 + ε) k , (1 − ε) k c2 c1

(28)

for every t. As 1 T

T

2π μk dt = , kT ρk2 (t)

0

for k sufficiently large, we have −2

(1 + ε)

μk

μ2k c1

2 −2 μk 4−γ 2π −2 (1 − ε) μk , kT c2

−2 4−γ

and hence, 2 γ

c1

kT (1 + ε)−2 2π

4−γ γ

2 γ

μk c2

kT (1 − ε)−2 2π

4−γ γ

.

This estimate together with (28) leads to 1 1 − ε γ4 −1 4−γ c1 c2 4 (1 + ε) γ

kT 2π

2 γ

ρk (t)

1 1 + ε γ4 −1 4−γ c2 c1 4 (1 − ε) γ

kT 2π

2 γ

.

Since ε is arbitrary, the proof of (25) is thus concluded. To prove (26) note that tan x˙ (t) = ρk (t)ϕ˙k (t) = μk . k ρk (t) Using the above estimates, we obtain 4

(1 − ε) γ (1 + ε)

8−γ γ

8−γ

c1

c24

1 γ (4−γ )

kT 2π

4 8−γ 1 2−γ 2−γ γ γ (4−γ ) c2 kT γ (1 + ε) γ μk . 8−γ 4 ρk (t) (1 − ε) γ 2π c1

Using again the arbitrariness of ε, (26) is proved, as well. The proof of (27) follows directly from (19). 2 Let us state the corresponding results for Eqs. (3) and (4). Corollary 4. Assume (H3), 0 < γ < 4, and e = 0. Then, the solutions xk (t) of Eq. (3) obtained in Corollary 1 satisfy the same estimates as in Theorem 5.

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Corollary 5. Assume (H3), d ∈ L∞ (R) and e = 0. Then, the solutions xk (t) of Eq. (4) obtained in Corollary 2 satisfy the same estimates as in Theorem 5, with γ = 3. Remark 1. Notice that, when e = 0, the estimates in Theorem 5 imply that lim x˙ktan (t) = +∞, if 0 < γ < 2, k→∞

lim x˙ktan (t) = 0,

if 2 < γ < 4,

k→∞

while |x˙ktan (t)| remains bounded, for large k, if γ = 2. It is also interesting to write the following particular case. Corollary 6. If e = 0 and there are γ ∈ ]0, 4[ and a constant c > 0 such that (H5)

lim r γ −1 h(t, r) = c

r→+∞

uniformly for almost every t, then the solutions xk (t) of Eq. (8) obtained in Corollary 3 satisfy the following estimates: lim

|xk (t)|

k→∞ kT ( 2π )

2 γ

|x˙ktan (t)|

lim

k→∞ kT ( 2π )

2−γ γ

1

= cγ ,

1

= cγ ,

and moreover, if 1 γ < 4, lim supx˙krad (t) e 1 k→∞

uniformly in t. Proof. It is sufficient to notice that (H5) corresponds to (H4), with c1 = c2 , and to apply Theorem 5. 2 As an immediate consequence, we have the following. Corollary 7. Assume e = 0, γ ∈ ]0, 4[ and that c(t) is constant with a positive value c. Then, the solutions xk (t) of Eq. (3) obtained in Corollary 1 satisfy the same estimates as in Corollary 6. Remark 2. When the function c is a positive constant and e is identically 0, the circular solutions of (3) with radius ρ > 0 have minimal period 2π γ τρ = √ ρ 2 c


3255

and tangential speed vρ =

√ 2−γ cρ 2 .

This situation is reflected by our estimates in Corollary 6. Indeed, the orbits of the largeamplitude solutions we have found are, in a sense, close to be circular, since the quotient between the maximal and minimal radius is close to 1. We have a similar situation for Eq. (4). Corollary 8. Assume e = 0, d ∈ L∞ (R) and that c(t) is constant with a positive value c. Then, the solutions xk (t) of Eq. (4) obtained in Corollary 2 satisfy the same estimates as in Corollary 6, with γ = 3. Analogous type of results can be deduced from Lemma 7 when e < 0, distinguishing the three cases 0 < γ < 1, γ = 1 and 1 < γ < 4. For briefness, let us only state the following. Theorem 6. Assume (H4), with 1 < γ < 4. If e < 0, then the solutions xk (t) of Eq. (8) obtained in Corollary 3 satisfy the following estimates: 2 |xk (t)| T = − e, k→∞ k 2 2π lim

|x˙ktan (t)| T = − e, k→∞ k 2π rad lim supx˙ (t) e 1 lim

k→∞

k

uniformly in t. Remark 3. As in Remark 2, we can compute explicitly the minimal period and the tangential speed of the circular solutions of (3) with radius ρ > 0, when the function c is a positive constant and the function e is constant with value e < 0, thus obtaining 2πρ 2

τρ = , cρ 4−γ − eρ 3

vρ =

cρ 4−γ − eρ 3 . ρ

This situation is well reflected by Theorem 6. 4. Multiplicity results In this section, we will prove the existence of solutions making a given number of revolutions around the origin. We first consider the general equation (8), and state the following. Theorem 7. Let e(t) be continuous, satisfying (H0), with minimal period T . Let h(t, r) be continuous, and assume (H1) and

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lim h(t, r) = 0

(H6)

r→+∞

uniformly for every t. Then, for every integer m 1, there exists km 1 such that, for every integer k km , if k and m are relatively prime, Eq. (8) has a periodic solution xk,m (t) with minimal period kT , which makes exactly m revolutions around the origin in the period time kT . Moreover, lim min|xk,m | = +∞

k→∞

and min|xk,m | = 1. k→∞ max|xk,m | lim

Proof. Following the proof of Theorem 4, if k 2πm/θ , there is a solution x(t) of (8) such that x(t + T ) = e

2πm k i

x(t),

for every t. We denote by xk,m (t) such a solution, and by ρk,m (t), ϕk,m (t) and μk,m its polar coordinates and angular momentum, respectively. Such a solution is kT -periodic and rotates exactly m times around the origin when t varies from 0 to kT . Recall that ρk,m (t) is T -periodic and, as in (16), lim μk,m = +∞.

k→∞

Moreover, as in (17) and (18), lim (min ρk,m ) = +∞,

k→∞

lim

k→∞

min ρk,m = 1. max ρk,m

Let us show that, if k is large enough and relatively prime with m, then kT is the minimal pe riod of xk,m (t). If not, xk,m (t) has minimal period kT m , for some integer m 2 which divides m, m kT thus making m revolutions in the period time m . Hence, also ρk,m (t) has period kT m . Let k = pm + q, for some integers p, q with q < m . Then, ρk,m (t) = ρk,m

kT t+ m

= ρk,m

qT t + pT + m

qT = ρk,m t + , m

for every t, thus showing that qT m is a period of ρk,m (t). Since period T , there is t˜ ∈ [0, T ] such that qT e t˜ + = e(t˜). m

qT m

< T and e(t) has minimal


3257

By (H6), there is rˆ > 0 such that r rˆ

⇒

h(t, r) 1 e t˜ + qT − e(t˜), 3 m

for every t ∈ [0, T ].

By the above, if k is large, then ρk,m (t) rˆ , for every t. From the first equation in (S) we deduce that e t˜ +

qT m

qT qT qT μ2 ˜ ˜ ˜ − h t + , ρk,m t + = ρ¨k,m t + − 3 (t˜ + m m m ρk,m = ρ¨k,m (t˜) −

qT m

)

μ2 3 (t˜) ρk,m

= e(t˜) − h t˜, ρk,m (t˜) . Consequently, e t˜ + qT − e(t˜) = h t˜ + qT , ρk,m t˜ + qT ˜ ˜ − h t , ρk,m (t ) m m m qT qT + h t˜, ρk,m (t˜) h t˜ + , ρk,m t˜ + m m 2 qT e t˜ + − e(t˜), 3 m and we have got a contradiction. Hence, xk,m (t) must have minimal period kT .

2

Remark 4. The estimates on the solutions made in Section 3, in the case when (H4) is satisfied, can be carried out in the same way for the solutions xk,m (t). Since these solutions make m revolutions around the origin in the time kT , the analogous of Theorems 5, 6 and their corollaries hold, simply replacing 2π by 2mπ . For instance, the analogous of (25) is 1 γ4 −1 1 |xk,m (t)| |xk,m (t)| γ4 −1 4−γ c1 c2 4−γ lim inf lim sup c2 c1 . 2 2 k→∞ kT γ k→∞ ( kT ) γ ( 2mπ ) 2mπ

Let us state now our multiplicity result. Theorem 8. Let e(t) be continuous, satisfying (H0), with minimal period T . Let h(t, r) be continuous, and assume (H1) and (H6). Then, for every n ∈ N, there is k(n) 1 such that, for every prime integer k k(n), Eq. (8) has at least n geometrically distinct periodic solutions with minimal period kT . Proof. Given n 1, let p1 , p2 , . . . , pn be the first n prime numbers. Correspondingly, let kp1 , kp2 , . . . , kpn be as in Theorem 7. Define k(n) = max{kp1 , kp2 , . . . , kpn }.

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By Theorem 7, for every prime number k k(n), Eq. (8) has periodic solutions xk,p1 , xk,p2 , . . . , xk,pn , with minimal period kT , which make exactly p1 , p2 , . . . , pn rotations around the origin, respectively, in the period time kT . 2 The following corollaries are immediate consequences of the above theorems. Corollary 9. Let c(t) and e(t) be continuous, satisfying (H0), (H3), and assume 1 < γ < 4. If e(t) has minimal period T , then the conclusions of Theorems 7 and 8 hold for Eq. (3). Corollary 10. Let c(t), d(t) and e(t) be continuous, assume (H0) and (H3). If e(t) has minimal period T , then the conclusions of Theorems 7 and 8 hold for Eq. (4). Let us concentrate now on Eq. (3). We have considered above the case when e(t) has minimal period T . We consider now the case when e(t) is constant but c(t) has minimal period T . Theorem 9. Assume e(t) is constant, with e(t) ≡ e 0, and 1 < γ < 4. If c(t) satisfies (H3) and has minimal period T , then the same conclusions of Theorems 7 and 8 hold for Eq. (3). Proof. Repeating the proof of Theorem 7, by contradiction we see that ρk,m (t) has to be periodic, and qT m < T . From the first equation in (S), γ −1

γ −4

qT m

-

γ −1

c(t) = −ρk,m (t)ρ¨k,m (t) + ρk,m (t)μ2 − ρk,m (t)e, so that c(t) has to be

qT m

-periodic, as well, which is a contradiction.

2

5. The case e > 0 In this section we will show that assumption (H0) is essential in order to have large amplitude periodic solutions to Eq. (8). This fact can easily be seen for the circular solutions of Eq. (3), when the functions c(t) and e(t) are constant, with c positive, and γ > 1. In that case, if e > 0, the circular solutions are confined in a compact region and their periods tend to infinity as they approach the boundary of that region. In the general case, we will show that, if e > 0 and the function h(t, r) tends to zero when r goes to infinity, periodic solutions of arbitrary large amplitude cannot exist. Theorem 10. Assume e > 0, and that (H6) holds. Then, every solution of (8) which attains a large enough distance from the origin is unbounded. More precisely, there is R > 0 such that, if x(t) is a solution of (8) with |x(t0 )| R for some t0 ∈ R, then the orbit of x(t) is unbounded. Proof. By (H6), there is r1 > 0 such that r r1

⇒

h(t, r) e , 2

for a.e. t ∈ R.

Choose R such that R > r1 +

e 21 5 17 + T e 1 + T 2 e. e 2 16

(29)


3259

Consider the equivalent system (S), and let ρ(t0 ) = |x(t0 )| R. We first assume ρ(t ˙ 0 ) 0, and prove by induction the following proposition: (Pn ) For every t ∈ [t0 + nT , t0 + (n + 1)T ], ρ(t) ˙

n−1 T e − e 1 , 2

ρ(t) R − (n + 1)T e 1 +

n2 − 3n − 2 2 T e, 4

and, moreover, n+1 ρ˙ t0 + (n + 1)T T e, 2 (n − 2)(n + 1) 2 T e. ρ t0 + (n + 1)T R − (n + 1)T e 1 + 4 We check that (P0 ) is true. First we show that for every t ∈ [t0 , t0 + T ].

ρ(t) > r1 ,

(30)

By contradiction, assume that there is t1 ∈ [t0 , t0 + T ] such that ρ(t1 ) = r1 and ρ(t) > r1 for every t ∈ [t0 , t1 [. Then, for t ∈ [t0 , t1 [, e ρ(t) ¨ e(t) − , 2 so that t ρ(t) ˙ = ρ(t ˙ 0) +

ρ(s) ¨ ds − e 1 −

Te , 2

t0

and hence t ρ(t) = ρ(t0 ) +

Te . ρ(s) ˙ ds R − T e 1 + 2

t0

On the other hand, by the choice of R,

e 21 3 Te 9 R − T e 1 + r1 + + T e 1 + T 2 e, 2 e 2 16 thus contradicting the fact that ρ(t1 ) = r1 .

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Having proved that (30) holds, we have that all the above inequalities hold for every t ∈ [t0 , t0 + T ], and hence the first two inequalities in (P0 ) hold true. Moreover, t 0 +T

t 0 +T

ρ(t ˙ 0 + T ) = ρ(t ˙ 0) +

ρ(s) ¨ ds t0

Te e e(s) − ds = 2 2

t0

and t 0 +T

ρ(t0 + T ) = ρ(t0 ) +

Te ρ(s) ˙ ds R − T e 1 + . 2

t0

Hence, (P0 ) has been proved. Now assume that (Pn ) holds, for some n ∈ N. First we show that ρ(t) > r1 ,

for every t ∈ t0 + (n + 1)T , t0 + (n + 2)T .

(31)

By contradiction, assume that there is t1 ∈ [t0 + (n + 1)T , t0 + (n + 2)T ] such that ρ(t1 ) = r1 and ρ(t) > r1 for every t ∈ [t0 + (n + 1)T , t1 [. Then, for t ∈ [t0 + (n + 1)T , t1 [, e ρ(t) ¨ e(t) − , 2 so that ρ(t) ˙ = ρ˙ t0 + (n + 1)T +

t ρ(s) ¨ ds

t0 +(n+1)T

n+1 Te n T e − e 1 − = T e − e 1 , 2 2 2

and hence ρ(t) = ρ t0 + (n + 1)T +

t ρ(s) ˙ ds

t0 +(n+1)T

(n − 2)(n + 1) 2 T e+ 4 n + t − t0 + (n + 1)T T e − e 1 2

R − (n + 1)T e 1 +

(n − 2)(n + 1) 2 Te T e − T e 1 + > R − (n + 1)T e 1 + 4 2 = R − (n + 2)T e 1 +

(n + 1)2 − 3(n + 1) − 2 2 T e. 4


3261

The function (x + 1)2 − 3(x + 1) − 2 2 T e 4

x → R − (x + 2)T e 1 + has as minimum value R−

e 21 5 17 − T e 1 − T 2 e, e 2 16

and this, together with (29), contradicts the fact that ρ(t1 ) = r1 . Having proved that (31) holds, we have that all the above inequalities hold for every t ∈ [t0 + (n + 1)T , t0 + (n + 2)T ], and hence the first two inequalities in (Pn+1 ) hold true. Moreover, ρ˙ t0 + (n + 2)T = ρ˙ t0 + (n + 1)T +

t0 +(n+2)T

ρ(s) ¨ ds

t0 +(n+1)T

n+1 Te+ 2

t0 +(n+2)T

e(s) −

t0 +(n+1)T

=

e ds 2

n+2 Te 2

and ρ t0 + (n + 2)T = ρ t0 + (n + 1)T +

t0 +(n+2)T

ρ(s) ˙ ds

t0 +(n+1)T

(n − 2)(n + 1) 2 n T e+T T e − e 1 R − (n + 1)T e 1 + 4 2 = R − (n + 2)T e 1 +

(n − 1)(n + 2) 2 T e. 4

Hence, (Pn+1 ) has been proved. Having proved that (Pn ) holds for every n ∈ N, as an immediate consequence we have that lim ρ(t) = lim ρ(t) ˙ = +∞.

t→+∞

t→+∞

Assume now that ρ(t ˙ 0 ) < 0. Setting η(t) = ρ(−t), we have that η verifies η¨ −

μ2 + h(−t, η) = e(−t). η3

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Moreover, η(−t0 ) > R and η(−t ˙ 0 ) > 0. So, the same arguments used above for ρ can now be used for η, thus leading to lim η(t) = lim η(t) ˙ = +∞,

t→+∞

t→+∞

which means lim ρ(t) = +∞,

t→−∞

The proof is thus completed.

lim ρ(t) ˙ = −∞.

t→−∞

2

As a direct consequence, we get the following two corollaries concerning Eqs. (3) and (4). Corollary 11. Assume e > 0, c ∈ L∞ (R), and γ > 1. Then, the same conclusion of Theorem 10 holds for Eq. (3). Corollary 12. Assume e > 0 and c, d ∈ L∞ (R). Then, the same conclusion of Theorem 10 holds for Eq. (4). Remark 5. Notice that in the proof of Theorem 10 we do not need the function h(t, r) to be periodic in the first variable t. In particular, in the above corollaries, c(t) and d(t) could be arbitrary bounded functions. Appendix A. Planar orbits The orbits of the solutions to a general system like (1) are planar. For N = 3, this fact is well known and can be found, e.g., in [6]. For the reader’s convenience, we give here a simple proof valid for any dimension N 3. Let x(t) be such a solution, defined on an interval I , with x(t) = x1 (t), . . . , xN (t) ∈ RN . We claim that the functions t → xi (t)x˙j (t) − xj (t)x˙i (t), with i, j ∈ {1, . . . , N}, are constant on I . Indeed, d (xi x˙j − xj x˙i ) = x˙i x˙j + xi x¨j − x˙j x˙i − xj xï = xi f t, |x| xj − xj f t, |x| xi = 0. dt ˙ 0 ) are linearly independent, for some t0 ∈ I , and let P(t0 ) be the plane Assume that x(t0 ) and x(t generated by these two vectors. Up to a permutation of the coordinates, we may assume, for instance, that x1 (t0 )x˙2 (t0 ) − x2 (t0 )x˙1 (t0 ) = 0.


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Then, x1 (t)x˙2 (t) − x2 (t)x˙1 (t) = 0 for every t ∈ I . In particular, x(t) and x(t) ˙ are linearly independent and generate a plane, which we denote by P(t). Writing the equations of this plane, we have that a vector v = (v1 , . . . , vN ) belongs to P(t) if and only if ⎧ x3 (t)x˙2 (t) − x2 (t)x˙3 (t) x1 (t)x˙3 (t) − x3 (t)x˙1 (t) ⎪ v3 = v1 + v2 , ⎪ ⎪ ⎪ x (t) x ˙ (t) − x (t) x ˙ (t) x 1 2 2 1 1 (t)x˙ 2 (t) − x2 (t)x˙ 1 (t) ⎪ ⎨ .. . ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ vN = xN (t)x˙2 (t) − x2 (t)x˙N (t) v1 + x1 (t)x˙N (t) − xN (t)x˙1 (t) v2 . x1 (t)x˙2 (t) − x2 (t)x˙1 (t) x1 (t)x˙2 (t) − x2 (t)x˙1 (t) Since all the coefficients are constant in t, we deduce that P(t) does not depend on t. Hence, the whole orbit of x(t) is contained in the plane P(t0 ). ˙ 0 ) are linearly dependent, for some t0 ∈ I . In this case the orbit Assume now that x(t0 ) and x(t must be contained in the straight line joining x(t0 ) with the origin. Otherwise, for some t1 ∈ I , ˙ 1 ) would have to be linearly independent and, by the previous arguments, the vectors x(t1 ) and x(t this would lead to a contradiction. References [1] S. Adachi, K. Tanaka, M. Terui, A remark on periodic solutions of singular Hamiltonian systems, NoDEA Nonlinear Differential Equations Appl. 12 (3) (2005) 265–274. [2] A. Ambrosetti, V. Coti Zelati, Critical points with lack of compactness and singular dynamical systems, Ann. Mat. Pura Appl. (4) 149 (1987) 237–259. [3] A. Ambrosetti, V. Coti Zelati, Noncollision orbits for a class of Keplerian-like potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (3) (1988) 287–295. [4] A. Ambrosetti, V. Coti Zelati, Perturbation of Hamiltonian systems with Keplerian potentials, Math. Z. 201 (2) (1989) 227–242. [5] A. Ambrosetti, V. Coti Zelati, Periodic Solutions of Singular Lagrangian Systems, Progr. Nonlinear Differential Equations Appl., vol. 10, Birkhäuser Boston Inc., Boston, MA, 1993. [6] V.I. Arnold, Mathematical Methods of Classical Mechanics, second ed., Grad. Texts in Math., vol. 60, Springer, New York, Heidelberg, 1978. [7] A. Bahri, P.H. Rabinowitz, A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal. 82 (2) (1989) 412–428. [8] D. Bonheure, C. De Coster, Forced singular oscillators and the method of lower and upper solutions, Topol. Methods Nonlinear Anal. 22 (2003) 297–317. [9] A. Capietto, J. Mawhin, F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc. 329 (1992) 41–72. [10] A. Capozzi, C. Greco, A. Salvatore, Lagrangian systems in the presence of singularities, Proc. Amer. Math. Soc. 102 (1) (1988) 125–130. [11] A. Capozzi, S. Solimini, S. Terracini, On a class of dynamical systems with singular potential, Nonlinear Anal. 16 (10) (1991) 805–815. [12] V. Coti Zelati, Periodic solutions for a class of planar, singular dynamical systems, J. Math. Pures Appl. (9) 68 (1) (1989) 109–119. [13] V. Coti Zelati, E. Serra, Collision and non-collision solutions for a class of Keplerian-like dynamical systems, Ann. Mat. Pura Appl. (4) 166 (1994) 343–362. [14] M. Degiovanni, F. Giannoni, A. Marino, Dynamical systems with Newtonian type potentials, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 81 (3) (1987) 271–277, (1988). [15] M. Degiovanni, F. Giannoni, Nonautonomous perturbations of Newtonian potentials, in: Nonlinear Variational Problems, vol. II, Isola d’Elba, 1986, in: Pitman Res. Notes Math. Ser., vol. 193, Longman Sci. Tech., Harlow, 1989, pp. 187–195. [16] M. del Pino, R. Manasevich, A. Montero, T -periodic solutions for some second order differential equations with singularities, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992) 231–243.

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[17] A. Fonda, Periodic solutions of scalar second order differential equations with a singularity, Acad. Roy. Belg. Cl. Sci. Mem. (3) IV (1993). [18] W.B. Gordon, Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc. 204 (1975) 113– 135. [19] C. Greco, Periodic solutions of a class of singular Hamiltonian systems, Nonlinear Anal. 12 (3) (1988) 259–269. [20] C. Greco, Existence of forced oscillations for some singular dynamical systems, Differential Integral Equations 3 (1) (1990) 93–101. [21] P. Habets, L. Sanchez, Periodic solutions of dissipative dynamical systems with singular potentials, Differential Integral Equations 3 (6) (1990) 1139–1149. [22] P. Habets, L. Sanchez, Periodic solutions of some Liénard equations with singularities, Proc. Amer. Math. Soc. 109 (4) (1990) 1035–1044. [23] A.C. Lazer, S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc. 99 (1) (1987) 109–114. [24] T. Levi-Civita, Fondamenti di Meccanica Relativistica, Zanichelli, Bologna, 1928. [25] P. Majer, Ljusternik–Schnirelman theory with local Palais–Smale condition and singular dynamical systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (5) (1991) 459–476. [26] P.H. Rabinowitz, Periodic solutions for some forced singular Hamiltonian systems, in: Analysis et cetera, Academic Press, Boston, MA, 1990, pp. 521–544. [27] M. Ramos, S. Terracini, Noncollision periodic solutions to some singular dynamical systems with very weak forces, J. Differential Equations 118 (1) (1995) 121–152. [28] E. Serra, S. Terracini, Noncollision solutions to some singular minimization problems with Keplerian-like potentials, Nonlinear Anal. 22 (1) (1994) 45–62. [29] K. Tanaka, Noncollision solutions for a second order singular Hamiltonian system with weak force, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (2) (1993) 215–238. [30] P. Torres, Existence of one-signed periodic solutions of some second order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations 190 (2003) 643–662. [31] P. Torres, Non-collision periodic solutions of forced dynamical systems with weak singularities, Discrete Contin. Dyn. Syst. 11 (2004) 693–698. [32] E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. 1, Springer, New York, Heidelberg, 1986.